GENUS DISTRIBUTIONS OF STAR-LADDERS
YICHAO CHEN, JONATHAN L. GROSS, AND TOUFIK MANSOUR
Abstract. Star-ladder graphs were introduced by Gross in his development of a quadratic-time algorithm for the genus distribution of a cubic outerplanar graph. This paper derives a formula for the genus distribution of star-ladder graphs, using Mohar’s overlap matrix and Chebyshev polynomials.
Newly developed methods have led to a number of recent papers that de- rive genus distributions and total embedding distributions for various families of graphs. Our focus here is on a family of graphs called star-ladders.
1. Introduction Genus distributions problems have frequently been investigated in the past quarter century, since the topic was inaugurated by Gross and Furst [6]. The contributions include [1,5,8,9, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 25, 26] and [27]. Gross [11] presents a quadratic-time algorithm for computing the genus distribution of any cubic outerplanar graph. He analyzes the structure of any cu- bic outerplanar graph and finds that such a graph can be obtained by a series of iterated edge-amalgamations of a new classes of graphs called star-ladders, so as to form a tree of star-ladders. Thus, beyond the direct interest in a closed formula for the genus distribution of star-ladders, such a formula is possibly a step toward a closed formula for the genus distribution of the cubic outerplanar graphs. Our closed formula in this paper for the genus distribution of star-ladders is derived with the aid of Mohar’s overlap matrices [17].
1.1. Star-ladders. An n-rung closed-end ladder Ln can be obtained by tak- ing the graphical cartesian product of an n-vertex path with the complete graph K2, and then doubling both its end edges. The new rungs obtained thereby are called end-rungs. Figure1 presents a 4-rung closed-end ladder. In [5], Furst, Gross, and Statman obtained a closed formula for the genus distribution of closed- end ladders.
2000 Mathematics Subject Classification. Primary: 05C10; Secondary: 30B70, 42C05. Key words and phrases. graph embedding; genus distribution; star-ladders; overlap matrix; Chebyshev polynomials. The work of the first author was partially supported by NNSFC under Grant No. 10901048.
1 2 YICHAO CHEN, JONATHAN L. GROSS, AND TOUFIK MANSOUR
Figure 1. The 4-rung closed-end ladder L4.
For an k-tuple of non-negative integers U = (n1, n2, . . . , nk) the star-ladder with signature U is the graph SLn1,n2,...,nk obtained from the cycle graph C2k, with consecutive edges labeled e1, e2, . . . , e2k as follows:
(1) For each i ≤ k such that ni = 0, double the edge e2i.
(2) For each of the ladders Ln1 ,Ln2 ,...,Lnk such that ni > 0,
• subdivide one end rung of Lni into three parts, and take the middle third as the root-edge;
• amalgamate Lni across its newly created root edge to the edge e2i.
The star-ladder SL2,1,0 is shown in Figure2.
Figure 2. The star-ladder SL2,1,0.
1.2. Genus polynomial. It is assumed that the reader is somewhat familiar with the basics of topological graph theory, as found in Gross and Tucker [7]. All graphs considered in this paper are connected. A graph G = (V (G),E(G)) is permitted to have both loops and multiple edges. A surface is a compact 2-manifold without boundary. In topology, surfaces are classified into the orientable surfaces Sg, with g handles (g ≥ 0), and the nonorientable surfaces Nk, with k crosscaps (k > 0). A graph embedding into a surface means a cellular embedding. For any spanning tree of G, the number of co-tree edges is called the Betti number of G, and is denoted by β(G). A rotation at a vertex v of a graph G is a cyclic order of all edge-ends (or equivalently, half-edges) incident with v.A pure rotation system ρ of a graph G is the collection of rotations at all vertices of G. An embedding of G into an oriented surface S induces a pure rotation system as follows: the rotation at v is the cyclic permutation corresponding to the order in which the edge- ends are traversed in an orientation-preserving tour around v. Conversely, by the GENUS DISTRIBUTIONS OF STAR-LADDERS 3
Heffter-Edmonds principle, every rotation system induces a unique embedding (up to homeomorphism) of G into some orientable surface S. The bijection of this correspondence implies that the total number of orientable embeddings is Y (dv − 1)!, v∈V (G) where dv is the degree of vertex v. A general rotation system is a pair (ρ, λ), where ρ is a pure rotation system and λ is a mapping E(G) → {0, 1}. The edge e is said to be twisted (respectively, untwisted) if λ(e) = 1 (respectively, λ(e) = 0). It is well-known that every oriented embedding of a graph G can be described by a general rotation system (ρ, λ) with λ(e) = 0 for all e ∈ E(G). By allowing λ to take non-zero values, we can describe the nonorientable embeddings of G. For any spaning tree T , a T - rotation system (ρ, λ) of G is a general rotation system (ρ, λ) such that λ(e) = 0, for all e ∈ E(T ). By the genus polynomial of a graph G, we mean the polynomial ∞ X i ΓG(z) = gi(G)z , i=0 where gi(G) means the number of embeddings of G into the orientable surface Si, for i ≥ 0.
1.3. Overlap matrices. Mohar [17] introduced an invariant that has subse- quently been used numerous times (e.g., [2,3,4]) in the calculation of distri- butions of graph embeddings, including non-orientable embeddings. We use Mo- har’s invariant here in our derivation of a formula for the genus distribution of star-ladders. Let T be a spanning tree of a graph G and let (ρ, λ) be a T -rotation system. Let e1, e2, . . . , eβ(G) be the cotree edges of T , where β(G) is the cycle rank of G. The overlap matrix of (ρ, λ) is the β(G) × β(G) matrix M = [mij] over Z2 such that 1, if i = j and e is twisted; i 1, if i 6= j and the restriction of the underlying pure m = ij rotation system to the subgraph T + e + e is nonplanar; i j 0, otherwise. When the restriction of the underlying pure rotation system to the subgraph T + ei + ej is nonplanar, we say that edges ei and ej overlap. The importance of overlap matrices is indicated by this theorem of Mohar [17]: 4 YICHAO CHEN, JONATHAN L. GROSS, AND TOUFIK MANSOUR
Theorem 1.1. Let (ρ, λ) be a general rotation system for a graph. Then the rank of any overlap matrix M for the corresponding embedding equals twice the genus of the embedding surface, if that surface is orientable, and it equals the crosscap number otherwise. The rank is independent of the choice of a spanning tree.
For drawing a planar representation of a rotation system on a cubic graph, we adopt the graphic convention introduced by Gustin [13], and used extensively by Ringel and Youngs (see [21]) in their solution to the Heawood map-coloring problem. There are two possible cyclic orderings of each trivalent vertex. Under this convention, we color a vertex black, if the rotation of the edge-ends incident on it is clockwise, and we color it white if the rotation is counterclockwise. We call any drawing of a graph that uses this convention to indicate a rotation system a Gustin representation of that rotation system. The approach here is similar approach to that used for ladders in [5]. In a Gustin representation of a rotation system for a graph, an edge is called matched if it has the same color at both endpoints; otherwise, it is called unmatched. In Figure3, we have indicated our choice of a spanning tree for a generic ladder Ln−1 by thicker lines and a partial choice of rotations at the vertices.
b1 b2 bn-2 bn-1
a1 a2 a3 an-1 an
Figure 3. A spanning tree and some rotations for the ladder Ln−1.
The following proposition facilitates the calculation of an overlap matrix for a ladder graph. The proof is simply to apply the Heffter-Edmonds face-tracing algorithm.
Proposition 1.2. In the ladder Ln−1, we choose all of the edges on one side of the ladder plus all of the rungs, except for the two created by doubling, as the edges of a spanning tree. We label the cotree edges a1, . . . , an, from one end of the ladder to the other, and we label the tree rungs b1, . . . , bn, from one end of the ladder to the other (as shown in Figure3). Then two cotree edges ai and ai+1, with 1 ≤ i ≤ n − 1, overlap if and only if the rung edge bi is unmatched. GENUS DISTRIBUTIONS OF STAR-LADDERS 5
By Proposition 1.2, the overlap matrix Mn of Ln−1 can be written as
x1 y1 y1 x2 y2 0 y x y X,Y 2 3 3 Mn = Mn = . . . , ...... 0 yn−2 xn−1 yn−1 yn−1 xn n n−1 where X = (x1, x2, . . . , xn) ∈ Z2 and Y = (y1, y2, . . . , yn−1) ∈ Z2 . Our notation X,Y Mn indicates not only that this is a tridiagonal n × n matrix, but also that the diagonal arrays just below and just above the main diagonal are identical. We say that such a matrix is symmetrically tridiagonal.
X,Y Corollary 1.3. Each symmetrically tridiagonal matrix Mn corresponds to ex- n−1 actly 2 different T -rotation systems for the ladder Ln−1, where T is the span- ning tree of Ln−1 given in Figure3. Proof. According to Proposition 1.2, changing the rotations at both endpoints of any or all of the rungs bj does not change any of the coefficients in the overlap matrix. Moreover, any other change of rotations in ρ does change the overlap matrix.
1.4. The rank-distribution polynomial. We now consider the set X,Y n n−1 An = {Mn | X ∈ Z2 and Y ∈ Z2 }, of all symmetrically tridiagonal n × n matrices over Z2. We define the rank- distribution polynomial of the set An to be the polynomial n X j Pn(z) = Cn(j)z , j=0 where Cn(j) is the number of different assignments of the variables xi and yk, X,Y with 1 ≤ i ≤ n and 1 ≤ k ≤ n − 1, for which the matrix Mn in An has rank j. Similarly, we consider the set 0,Y n−1 On = {Mn | Y ∈ Z2 }, and we define the rank-distribution polynomial of On to be the polynomial n X j (1) On(z) = Bn(j)z , j=0 where Bn(j) is the number of different assignment of the variables y1, . . . , yn−1 for Y 0,Y which the matrix Mn = Mn in On has rank j. 6 YICHAO CHEN, JONATHAN L. GROSS, AND TOUFIK MANSOUR
We recall that the Chebyshev polynomials of the second kind are defined by
(2) Un(t) = 2tUn−1(t) − Un−2(t),U0(t) = 1,U1(t) = 2t.
Lemma 1.4. The rank-distribution polynomial On(z) for symmetrically tridiago- nal n × n matrices satisfies the recurrence relation 2 (3) On(z) = On−1(z) + 2z On−2(z) with the initial conditions 2 (4) O0(z) = O1(z) = 1 and O2(z) = z + 1. Moreover, √ n 1 1 1 (5) On(z) = (iz 2) Un √ + Un−2 √ 2iz 2 2 2iz 2 2 th where i = −1, and where Um is the m Chebyshev polynomial of the second kind.
Proof. It is directly ascertainable that the sequence of functions Bn(j) satisfies the recurrence system
B0(j) = 0 for j 6= 0
Bn(0) = 1 for all n = 0, 1,...
B2(2) = 1
(6) Bn(j) = Bn−1(j) + 2Bn−2(j − 2).
It follows, in turn, from its definition (1) that the polynomial On(z) satisfies the recursion (3) and the initial conditions (4). Applying induction on n to the recursion (3), while using the Chebyshev recursion (2), we obtain equation (5): √ n 1 1 1 On(z) = (iz 2) Un √ + Un−2 √ , 2iz 2 2 2iz 2 which completes the proof. Theorem 1.5. (Furst et al.[5]) The number of embeddings of the closed-end ladder Ln−1 into the orientable surface Si is 2n−2+i n−i 2n−3i , when i ≤ b n c g (L ) = i n−i 2 i n−1 0, otherwise.
Proof. Let the genus polynomial of the ladder Ln−1 be
X i ΓLn−1 (z) = gi(Ln−1)z . i≥0 GENUS DISTRIBUTIONS OF STAR-LADDERS 7
By Formula (5) of Lemma 1.4 and Corollary 1.3, we have n−1 ΓLn−1 (z) = 2 On(z) ( ) X n − j X n − 2 − j (7) = 2n−1 2j z2j − 2j z2j+2 . j j j≥0 j≥0 2j Note that gj(Ln−1) is equal to the coefficient of z . By (7), we have n − j n − j − 1 g (L ) = 2n−1 2j − 2j−1 . j n−1 j j − 1 n−m n−m n−m−1 By Newton’s identity m = m m−1 , the theorem follows.
2. Rank-Distribution Polynomial of Star-Ladders
We fix a spanning tree T of SLn1,n2,...,nk , shown by thicker lines in Figure4, with cotree edges e, e1,0, e1,1, . . . , e1,n1 , e2,0, e2,1, . . . , e2,n2 ,..., ek,0, ek,1, . . . , ek,nk , also as shown.
Property 2.1. The cotree edge e overlaps the cotree edge ei,0 if and only if the edge bi,0 is unmatched, for i = 1, 2, ··· , k.
Property 2.2. The cotree edges ei,ij and ei,ij +1 overlap if and only if the edge bi,ij +1 is unmatched, for i = 1, 2, ··· , k,ij = 0, 1, ··· , ni − 1.
b2,n +1 2 e 2,n2
b2,1 e2,0
b2,0 e e 1,n1 1,0
b b b b b b 1,n1+1 1,1 1,0 k,0 k,1 k,nk+1
e e e k,1 k,nk
Figure 4. A spanning tree for the star-ladder SLn1,n2,nk 8 YICHAO CHEN, JONATHAN L. GROSS, AND TOUFIK MANSOUR
Let W = (w , w , . . . , w ) ∈ n and Y = (y , y , . . . , y ) ∈ ki , for i = 1 2 k Z2 i i1 i2 iki Z2
1, 2, ··· , k. Then the overlap matrix of a star-ladder SLn1,n2,...,nk can be written in the following form: M W, Y1,Y2,··· ,Yk = n1,n2,...,nk 0 w1 0 0 ··· 0 w2 0 0 ··· 0 ··· wk 0 0 ··· 0 w1 0 y1,1 0 y1,1 0 y1,2 . . 0 y .. .. 1,2 . .. . . y1,n 1 0 y1,n 0 0 1 w2 0 0 y2,1 0 y2,1 0 y2,2 .. .. 0 y2,2 . . , . . . .. y 2,n2 0 y 0 2,n2 ...... . . . . wk 0 yk1 0 yk,1 0 yk,2 . . .. .. 0 yk,2 . . . .. . yk,nk
0 yk,nk 0 where wi = 1, for i = 1, 2, . . . , k, if and only if bi,0 is unmatched, and where yi,ik = 1, for i = 1, 2, . . . , k and ik = 1, 2, . . . , ni, if only if bi,ik is unmatched.
Proposition 2.3. For a fixed overlap matrix of the form M W, Y1,Y2,··· ,Yk , corre- n1,n2,...,nk sponding to the spanning tree T in a star-ladder graph SLn1,n2,...,nk , there are Pk (n +1) exactly 2 i=1 i different T -rotation systems corresponding to that matrix.
Proof. This proof is like that of Property 1.3.
We let Sn1,n2,...,nk denote the set of all matrices over Z2 that are of the type M W, Y1,Y2,··· ,Yk . We let D(j) denote the number of different assignments of the n1,n2,...,nk variables w , y for which the matrix M W, Y1,Y2,··· ,Yk in S has rank j, j i,ik n1,n2,...,nk n1,n2,...,nk where j = 1, 2, ··· , n; i = 1, 2, ··· , k; and ik = 1, 2, . . . , ni. Additionally, We define the rank-distribution polynomial
n1+n2+···+nk+k+1 X j (8) Sn1,n2,...,nk (z) = D(j)z . j=0 GENUS DISTRIBUTIONS OF STAR-LADDERS 9
Theorem 2.4. The rank-distribution polynomial Sn1,n2,...,nk (z) of the overlap ma- trices of a star-ladder graph SLn1,n2,...,nk satisfies the recurrence relation 2 Sn1,n2,...,nk (z) = Sn1,n2,...,nk−1(z) + 2z Sn1,n2,...,nk−2(z) with the initial conditions k−1 2 Sn1,n2,...,nk−1,0(z) = Sn1,n2,...,nk−1 (z) + 2 z On1+1(z)On2+1(z) ··· Onk−1+1(z) and 2 Sn1,n2,...,nk−1,1(z) = Sn1,n2,...,nk−1,0(z) + 2z Sn1,n2,...,nk−1 (z), where Om(z) is the rank distribution polynomial of the overlap matrices of the ladder graph Lm−1, as defined in Equation (1).
Proof. There are two cases.
Case 1. For ynk = 0. It is clear that rank(M W,Y1,Y2,··· ,Yk ) = rank(M W,Y1,Y2,··· ,Yk ) n1,n2,...,nk n1,n2,...,nk−1 so it contributes a term Sn1,n2,...,nk−1(z).
Case 2. For ynk = 1. If ynk−1 = 0, then rank(M W,Y1,Y2,··· ,Yk ) = 2 + rank(M W,Y1,Y2,··· ,Yk ). n1,n2,...,nk n1,n2,...,nk−2
Otherwise ynk−1 = 1, under which circumstance we add the last row and last column, respectively, to row n1+n2+...+nk +k and to column n1+n2+...+nk +k. We see thereby that rank(M W,Y1,Y2,··· ,Yk ) is equal to 2 plus the rank of the upper- n1,n2,...,nk left matrix, which has the form of M W,Y1,Y2,··· ,Yk , that is, n1,n2,...,nk−2 rank(M W,Y1,Y2,··· ,Yk ) = 2 + rank(M W,Y1,Y2,··· ,Yk ). n1,n2,...,nk n1,n2,...,nk−2 2 In total, it contributes a term 2z Sn1,n2,...,nk−2(z).
Hence, the polynomials Sn1,n2,...,nk (z) satisfy the recurrence relation 2 Sn1,n2,...,nk (z) = Sn1,n2,...,nk−1(z) + 2z Sn1,n2,...,nk−2(z), for all nk ≥ 2 and k ≥ 3. Note that for k = 2, the definition (8) implies that
n1+n2+3 X j Sn1,n2 (z) = D(j)z = On1+n2+2(z), j=0 where X n − j X n − 2 − j O (z) = 2j z2j − 2j z2j+2. n j j j≥0 j≥0 10 YICHAO CHEN, JONATHAN L. GROSS, AND TOUFIK MANSOUR
Moreover, for k = 3, Theorem 2.4 implies these three equations:
2 Sn1,n2,n3 (z) = Sn1,n2,n3−1(z) + 2z Sn1,n2,n3−2(z) 2 Sn1,n2,0(z) = On1+n2+2(z) + 4z On1+1(z)On2+1(z) 2 Sn1,n2,1(z) = Sn1,n2,0(z) + 2z On1+n2+2(z).
To solve the recursion of Theorem 2.4, we define
X n1 n2 nk (9) S(t1, t2, . . . , tk, z) = Sn1,n2,...,nk (z) t1 t2 ··· tk n1,n2,...,nk≥0 and X n (10) O(t, z) = On(z)t . n≥1
Rewriting the recurrence relation in the statement of Lemma 1.4 in terms of a generating function, we obtain
(1 + z2t)t (11) O(t, z) = . 1 − t − 2z2t2
Rewriting the recurrence in the statement of Theorem 2.4 as a generating function, we obtain
S(t1, t2, . . . , tk, z) = S(t1, t2, . . . , tk−1, z) k−1 k−1 2 Y −1 2 + 2 z tj O(tj, z) + 2z tkS(t1, t2, . . . , tk−1, z) j=1 2 2 + tkS(t1, t2, . . . , tk, z) + 2z tkS(t1, t2, . . . , tk, z), which, by (11), is equivalent to
1 + 2z2t S k S (t1, t2, . . . , tk, z) = 2 2 (t1, t2, . . . , tk−1, z)(12) 1 − tk − 2z tk k−1 2 Qk−1 2 2 z (1 + z tj) + j=1 k ≥ 3. Qk 2 2 j=1(1 − tj − 2z tj ) GENUS DISTRIBUTIONS OF STAR-LADDERS 11
Using the fact that Sn1,n2 (z) = On1+n2+2(z), we obtain
X n1 n2 S(t1, t2, z) = On1+n2+2(z)t1 t2 n1,n2≥0 X n−3 n−4 n−4 n−3 = On(z)(t1 + t1 t2 + ··· + t1t2 + t2 ) n≥2 X tn−2 − tn−2 = O (z) 1 2 n t − t n≥2 1 2
O(t1, z) − t1 O(t2, z) − t2 = 2 − 2 t1(t1 − t2) t2(t1 − t2) 2 2 2 2 2 (1 + 2z t1)(1 + 2z t2) + z (3 + 2z t1 + 2z t2) = 2 2 2 2 . (1 − t1 − 2z t1)(1 − t2 − 2z t2) which, by (11), implies (1 + 2z2t )(1 + 2z2t ) + z2(3 + 2z2t + 2z2t ) S 1 2 1 2 (13) (t1, t2, z) = 2 2 2 2 . (1 − t1 − 2z t1)(1 − t2 − 2z t2) Iterating (12) we obtain
k 2 Y 1 + 2z tj S(t , t , . . . , t , z) = S(t , t , z) 1 2 k 1 2 1 − t − 2z2t2 j=3 j j 2 Pk j−1 Qj−1 2 Qk 2 z 2 (1 + z ti) (1 + 2z ti) + j=3 i=1 i=j+1 , Qk 2 2 j=1(1 − tj − 2z tj ) which, by (13), implies the following result. Theorem 2.5. Let k ≥ 2. Then the rank distribution of the overlap matrices for the star-ladder graph S(n1,n2,...,nk) is given by the generating function Qk 2 j=1(1 + 2z tj) S(t1, t2, . . . , tk, z) = Qk 2 2 j=1(1 − tj − 2z tj ) 2 Pk j−1 Qj−1 2 Qk 2 z 2 (1 + z t`) (1 + 2z t`) + j=1 `=1 `=j+1 . Qk 2 2 j=1(1 − tj − 2z tj )
Now our aim is to find an explicit formula for Sn1,n2,...,nk (z) by finding the n n1 n2 nk coefficient of t = t1 t2 ··· tk in the generating function S(t1, t2, . . . , tk, z). At first, note that the coefficient of tn in 1 Qk 2 2 j=1(1 − tj − 2z tj ) 12 YICHAO CHEN, JONATHAN L. GROSS, AND TOUFIK MANSOUR
(see Lemma 1.4) is given by ! k 1 Y n 1 [tn] = [t j ] Qk 2 2 j 1 − t − 2z2t2 j=1(1 − tj − 2z tj ) j=1 j j k √ 1 Y nj = (i 2z) Unj √ j=1 2i 2z k √ Pk 1 j=1 nj Y (14) = (i 2z) Unj √ j=1 2i 2z and that k Y X |A| Y (15) (1 + utj) = u ta, j=s A⊆[s,k] a∈A for any k ≥ s, where [a, b] = {a, a + 1, . . . , b}. We define √ 1 nj −χA(j) ρA(nj) = (i 2z) Un −χ (j) √ , j A 2i 2z th where Un(t) is the n Chebyshev polynomial of the second kind, and χA(j) is defined to be 1 if j ∈ A or 0 otherwise, and i2 = −1. Now Theorem 2.5 together with (14) and (15) imply the following result.
Theorem 2.6. Let k ≥ 2, let n1, n2, . . . , nk ≥ 0. Then the rank-distribution n Sn1,n2,...,nk (z) = [t ]S(t1, t2, . . . , tk, z) is given by the polynomial k X 2 |A| Y Sn1,n2,...,nk (z) = (2z ) ρA(nj) A⊆[1,k] j=1 k k 2 X X X |B|+j−1 2|A|+2|B| Y + z 2 z ρA∪B(nj). j=1 A⊆[1,j−1] B⊆[j+1,k] j=1 Theorem 2.6 reveals the following nice property:
Corollary 2.7. For k ≥ 2, let π = (n1, n2, . . . , nk) be a n-tuple of k nonnegative 0 integers, and let π be any permutation of π. Then Sπ(z) = Sπ0 (z).
Theorem 2.8. The genus polynomial of the star-ladder SLU is as follows: Pk √ j=1(nj +1) ΓSLU (z) = 2 Sn1,n2,...,nk ( z), where Sn1,n2,...,nk (z) is the rank-distribution polynomial defined by equation (8).
Proof. The theorem follows from Proposition 2.3. GENUS DISTRIBUTIONS OF STAR-LADDERS 13
Example 2.9. Let k = 3 and let us find the polynomial S2,1,0(z). After evaluating each sum in the formula of S2,1,0(z) according to Theorem 2.6, we obtain 2 4 6 2 4 2 4 6 S2,1,0(z) = (1 + 7z + 12z + 4z ) + (2z + 6z ) + (4z + 16z + 12z ) = 1 + 13z2 + 34z4 + 16z6.
2 4 6 Thus, ΓSL2,1,0 (z) = 64S2,1,0(z) = 64 + 832z + 2176z + 1024z . Example 2.9 can be extended as follows. Let √ n 1 pn = (i 2z) Un √ 2i 2z with i2 = −1. Then Theorem 2.6 for k = 3 gives 2 Sa,b,c(z) = papbpc + 2z (pa−1pbpc + papb−1pc + papbpc−1) 4 6 + 4z (pa−1pb−1pc + pa−1pbpc−1 + papb−1pc−1) + 8z pa−1pb−1pc−1 2 4 4 6 + z papbpc + 2z papb−1pc + 2z papbpc−1 + 4z papb−1pc−1 2 4 4 6 + 2z papbpc + 4z papbpc−1 + 2z pa−1pbpc + 4z pa−1pbpc−1 2 4 6 + 4z papbpc + 4z (pa−1pbpc + papb−1pc) + 4z pa−1pb−1pc, which implies this formula 2 2 2 Sa,b,c(z) = (1 + 7z )papbpc + 2z (1 + 3z )(pa−1pbpc + papb−1pc + papbpc−1) 4 2 6 + 4z (1 + z )(pa−1pb−1pc + pa−1pbpc−1 + papb−1pc−1) + 8z pa−1pb−1pc−1. Example 2.10. Applying this formula for several values of a, b, c we obtain the following values: 2 2 4 S0,0,0(z) = 1 + 7z S1,0,0(z) = 1 + 9z + 6z 2 4 2 4 6 S2,0,0(z) = 1 + 11z + 20z S1,1,0(z) = 1 + 11z + 16z + 4z 2 4 6 2 4 6 S3,0,0(z) = 1 + 13z + 38z + 12z S2,1,0(z) = 1 + 13z + 34z + 16z 2 4 6 S1,1,1(z) = 1 + 13z + 30z + 20z .
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College of Mathematics and Econometrics, Hunan University, 410082 Chang- sha, China E-mail address: [email protected]
Department of Computer Science, Columbia University, New York, NY 10027 USA E-mail address: [email protected]
Department of Mathematics, University of Haifa, 31905 Haifa, Israel E-mail address: [email protected]